PowerPoint Presentation - Bernd Sturmfels Lecture 1math.sjtu.edu.cn/faculty/ykwu/data/Talk/20090220.pdf智识份子的五个条件 • Ability to think critically, communicate with

MA106 现代数学选讲

周五，7，8 (14:00-15:40)，东上院202

课程安排

http://math.sjtu.edu.cn/teacher/wuyk/2009.pdf

We hope that the appearance of these articles in a single place will encourage mathematicians to share with others their perceptions on this beautiful subject, or on other matters dealing with more directlythe humanistic side of our nature. –Raymond G. Ayoub, (Ed.) Musings of the Masters: An Anthology of Mathematical Reflections, MAA, 2004.

编者是华罗庚的第一个博士生

课程目标

大学的目标然而，在大学博览会，我们常听到有些大学的宣传，他们的大学目标以「如果你来我的学校念书，出去的时候一定保证有职业。」为口号来吸引学生，这个口号基本上不是大学的目标，而是职业学校的目标。所谓大学university或是college，它的目标应该是为社会先预备一些对社会有用的、能够批判、求改进的人。我想这个深度应该是大学而且是通识教育的目标。-- 黄昆岩http://www.fdcollege.fudan.edu.cn/new/detail_xwck.php?id=41

智识份子的五个条件• Ability to think critically, communicate with

precision, cogency and force;• Informed acquaintance with mathematical,

physical and biological science; to be life-long learners;

• Widen the vision; • Thinking about moral and ethical problems; • Achieve depth in one’s own field of knowledge.

--- Henry Rosovsky, The University: An Owner's Manual, W. W. Norton & Company, 1991.

The University的书评Perhaps the guiding force behind military education is an assumption that a single failure to learn may make or break a military operation, and very likely will cause needless casualties. This is in sharp contrast to, say, the “Harvard” model of education. At Harvard, indeed (one hopes) in the university in general, one may assume that students are independently motivated, that they are “burnin‘ to learn.” Or if they aren’t, they ought to be. Not so in the armed forces. Students may require external motivation, both positive and negative. -- http://www.amazon.com/University-Owners-Manual-Henry-Rosovsky/dp/0393307832

常态的教育这个建议的核心理念是一个什么东西呢？就是说国防教育就是平常的教育，就是常态的教育，简单地说就是战争时期的教育，不可能搞成那种短训班，也不可能搞成直接为战争服务的这种大学，那么当时中国自由主义知识分子，他为什么会有这样一个远见呢？主要是他们吸取了第一次世界大战欧美国家在战争当中，把一些青年学生送到了前线去。那么在战后，国家恢复的时候，整个的人才就非常地匮乏。这是当时自由主义知识分子吸取了西方一个教训，然后给政府提的一个建议。这个政府比较理智地也比较及时地吸收了自由主义知识分子，战时教育，就是平常教育的理念，在这种情况下，决定了当时要把中国北方和南方一些有名的大学来建立联合大学的一种模式。

-- 谢泳：知识分子退步了 西南联大的当代意义

http://phtv.ifeng.com/program/sjdjt/200810/1009_1613_823323_1.shtml

人类心智的荣耀在拿破仑统治的法国，数学只是用来加强法国军事目标的工具。法国数学家约瑟夫·傅立叶曾批评德国学校的教育忽视了那些实际问题，柏林大学的教授卡尔·雅克比在1830年写给巴黎的勒让德的信中，对这件事作了如下回应：

傅立叶先生认为数学的主要目的是公众的应用和解释自然现象，这是没错的。但是象他那样的哲学家也应该知道，科学的唯一目的是人类心智的荣耀。在这种观点之下，数论之中的问题与世界系统的问题应该是具有相同价值的。

-- 素数的音乐，马科斯·杜·索托伊著，孙维昆译，湖南科学技术出版社, 2007.

The Community of Scholars I``Those in power sense, rather mistakenly, that in the exercise of their mission, they have no need of scholars, but in fact they need technicians, or if you prefer, researchers in the modern sense of this term. ... Such is, throughout the world, the natural progression of thought of the leaders who, because of their individual experience, cannot but deny the scientific ideal.’’

The Community of Scholars II`` In the heritage of the classical ideal of science, there is an indisputable part without which there would be no living scientific community, but an organization of qualified workers who would quickly disappear. And across winds and seas, our community reaffirms that component that consists of loyalty in discussions, freedom in research and communication, indifference to material gain. But this heritage has become a heavy one since responsibilities toward the society of humans have emerged.’’ – Andre Lichnerowicz, The Community of Scholars, Musings of the Masters, (R.G. Ayoub, Ed.), pp. 185—198, MAA, 2004.

If science no longer appears as being capable of influencing people’s hearts, and as being indifferent to morality, could it not have a detrimental as well as a beneficial effect? And would it not make us lose sight of those things not pertaining to it; the love of truth is without doubt a fine thing but a vain aim if to pursue it we sacrifice goals infinitely more precious such as goodness, compassion, love of neighbor. -- Henri Poincaré (1854--1912)

数学是最简单，最明确，有广泛应用，适于培养思维的学科

数学为人类和社会提供了可靠的有效思维方式 — 归纳与演绎相结合的思维方式。归纳与演绎的思维方式本来是一般科学（不仅是自然科学）的思维方式，但是她在数学中具有最明确的形式，数学是她的最好的载体，而且可以说她是由数学研究而发生、发展的。前面的论述说明：我国和西方在文化传统的根本出发点、基本思维方式上是不同的。从希腊和西方的文化传统可以清楚看出：数学是关键点之一。而数学在我国固有的文化传统中是没有什么地位的，就是在现代，人们可能更多地还是将她看成是一门科学甚至工具。因此，为了我国的现代化和民族的振兴，急切需要在我国优秀文化传统的基础上，让数学融入中国文化传统。这是一项极端重要、伟大而又长期的艰巨任务。

严士健，让数学融入我国文化传统， 光明日报, 2007-05-08, http://www.gmw.cn/CONTENT/2007-05/08/content_602628.htm

英才教育在5%的英才之外，美国的教育``失败’’了。但是，这成功的5%，支撑了美国经济50余年在世界上的长盛不衰。

当我们的孩子在学校内外花费很多时间，去操练并不涉及数学本质的技巧时，法国的学校在踏踏实实地把数学本质的东西教给学生，一步一个脚印，把孩子们领上科学之路。

－ 张英伯，谈谈英才教育，中国数学会通讯，2008年第四期。

One sees the difficulties that are encountered by a young person when he tries to master the first semester of his mathematical lectures. A language is used there which he does not know. Once again he is in the position that every individual, as a child, goes through when he learns his mother tongue. Only through familiarity, and much determination, as well as imagination is it possible to master mathematics in all its intricacies. –Wilhelm Maak, Goethe and Mathematics, Musings of the Masters, (R.G. Ayoub, Ed.), pp. 231—246, MAA, 2004.

The enchanting charms of this sublime subject do not reveal themselves except to those who have the courage to plumb its depths. – Carl Friedrich Gauss (1777--1855)

Gröbner Bases

吴耀琨 （上海交通大学）

2009/2/20

We copy almost all slides from

Bernd Sturmfels, Gröbner Bases, math.berkeley.edu/~bernd/grobner.ppt

Gröbner Bases with Bernd Sturmfels I

We follow the video freely downloadable at the above address and you are encouraged to view that video.

http://www.msri.org/communications/vmath/VMathVideosSpecial/VideoSpecialInfo/3020/show_video

You can also go to Lecture 4 available here:

http://math.berkeley.edu/%7Elpachter/239/

• The series "New Horizons in Undergraduate Mathematics" will showcase great lecturers speaking on topics from current research that are both important, accessible and ready to enter the undergraduate curriculum.

• Today, the Gröbner bases theory has been very useful in providing computational tools to help solve a wide array of problems in Mathematics, Science, Engineering and Computer Science.http://www.msri.org/communications/vmath/special_productions/production1/index_html

Gröbner Bases with Bernd Sturmfels II

Gröbner Bases with Bernd Sturmfels III

Gröbner Bases• Method for computing with

multivariate polynomials• Generalizes well-known algorithms:

– Gaussian Elimination– Euclidean Algorithm (for computing gcd)– Simplex Algorithm (linear programming)– Sylvester Resultant (for eliminating one

variable from two polynomials)

Buchberger,広中平祐，Shirshov

The theory of Gröbner bases for polynomial rings was developed by Bruno Buchberger in 1965, who named them after his advisor Wolfgang Gröbner. The Association for Computing Machinery awarded him its 2007 Paris Kanellakis Theory and Practice Award for this work. An analogous concept for local ringswas developed independently by Heisuke Hironaka in 1964, who named them standard bases. The analogous theory for free Lie algebras was developed by A. I. Shirshov in 1962 but his work remained largely unknown outside the Soviet Union.

http://visualwikipedia.com/en/Gr%C3%B6bner_basis

Bruno Buchberger (1942-- )Wolfgang Gröbner (1899-1980)

Gröbner first studied engineering but switched tomathematics in 1929. After his promotion he did further studies in Göttingen under Emmy Noether, in what is now known as commutative algebra.

http://visualwikipedia.com/en/Wolfgang_Gr%C3%B6bner

• 1942: Born in Innsbruck, Austria • 1960-66: Study of mathematics, University of Innsbruck.

Secondary subjects: experimental physics, philosophy. • 1966: Ph.D. in mathematics. University of Innsbruck• Thesis: On Finding a Vector Space Basis of the Residue Class

Ring Modulo a Zero Dimensional Polynomial Ideal (German). • Thesis Advisor: Wolfgang Gröbner• 1987: Establish the Research Institute for Symbolic

Computation (RISC) in the Johannes Kepler University Linz, Austria.

Bruno Buchberger

Bernd Sturmfels held postdoctoral position at RISC after he received his

Ph.D. at the University of Washington in 1987

• Gröbner-Shirshov bases： an analogue of Gröbnerbases for free Lie algebras

• A. I. Shirshov, Some algorithm problems for Lie algebras. Sibirsk. Mat. Zh., 3 (1962), 292-296.

• L.A. Bokut, 陈裕群，Groebner-Shirshov Bases for Lie Algebras: after A. I. Shirshov, eprintarXiv:0804.1254

A. I. Shirshov Leonid Bokut Efim IsaakovichZelmanov (1994 Fields medalist)

広中平祐• I think I can enjoy much more than

nonmathematicians just by looking at nature from the point of view of numbers and additions and multiplications.

• The resolution of singularities was done by hand. It doesn’t use large theory or techniques. It’s done just by hand. Andrew Wiles put many things together for his proof. I was just making new definitions and working case by case. -- 広中平祐(1931-- ), 1970 Fields medalist

General Setup• Set of input polynomials F = {f1,…,fn}

• Set of output polynomials G = {g1,…,gm}

Information about F easier to understand (more visible) through inspection of G

Buchberger’s Algorithm

Gaussian EliminationExample:

2x+3y+4z = 5 & 3x+4y+5z = 2⇒ x = z-14 & y = 11-2z

In Gröbner bases notation

Input: F = {2x+3y+4z-5, 3x+4y+5z-2}Output: G = {x-z+14, y+2z-11}

Euclidean Algorithm• Computes the greatest common divisor of

two polynomials in one variable.• Example: f1 = x4-12x3+49x2-78x+4, • f2 = x5-5x4+5x3+5x2-6x

gcd(f1,f2) = x2-3x+2

In Gröbner bases notationInput: F = {x4-12x3+49x2-78x+40, x5-5x4+5x3+5x2-6x}Output: G = {x2-3x+2}

Computinggcd of

integers is a special

case of it.

Integer ProgrammingMinimize the linear function

P+N+D+QSubject to P,N,D,Q > 0 integer and

P+5N+10D+25Q = 117This problem has the unique solution

(P,N,D,Q) = (2,1,1,4)

• Penny: 一便士

• Nickle: 五分镍币

• Dime: 一角银币

• Quarter:二角五分硬币

Integer Programming and Gröbner Bases

• Represent a collection C of coins by a monomial panbdcqd in the variables p,n,d,q.– E.g., 2 pennies and 4 dimes is p2d4

• Input set F = {p5-n, p10-d, p25-q}– Represents the basic relationships among coins

• Output set G = {p5-n, n2-d, d2n-q, d3-nq}– Expresses a more useful set of replacement rules.

E.g., the expression d3-nq translates to: replace 3 dimes with a nickel and a quarter

Integer Programming (cont’d)

• Given a collection C of coins, we use rules encoded by G to transform (in any order) Cinto a set of coins C’ with equal monetary value but smaller number of elements

• Example (solving previous integer program): p17n10d5 p12n11d5 . . . p2n13d5

p2ndq4 . . . p2n13dq2 p2n12d3q

Integer Programming (cont’d)

• Gröbner Bases give a method of transforming a feasible solution using local moves into a global optimum.

• This transformation is analogous to running the Simplex Algorithm

• Now, the general theory ….

Polynomial Ideals• Let F be a set of polynomials in K[x1,…,xn].• Here K is a field, e.g. the rationals Q, the

real numbers R, or the complex numbers C.• The ideal generated by F is

<F> = { p1f1+ ··· + prfr | fi ∈ F, pi ∈ K[x] }These are all the polynomial linear combinations of elements in F.

• F is an ideal itself iff <F>=F.

Examples of IdealsFor each example we have seen,

<F> = <G>– <{2x+3y+4z-5, 3x+4y+5z-2}> = <{x-z+14, y+z-11}>– <{x4-12x3+49x2-78x+40, x5-5x4+5x3+5x2-6x}>

= <{x2-3x+2}> – <{p5-n, p10-d, p25-q}> = <{p5-n, n2-d, d2n-q, d3-nq}>

In each example, the polynomial consequences for each set (i.e. the ideal generated by them) are the same, but the elements of G reveal more structure than those of F.

Ideal Equality• How to check that two ideals <F> and <G> are

equal?– need to show that each element of F is in <G> and

each element of G is in <F>• Coin example: d3-nq = n(p25-q) - (p20+p10d+d2)(p10-d)

+ p25(p5-n)标准型主义： to check if they have the same

`normal form’ of generating sets. The normal forms here are Reduced Gröbner bases!

Multivariate Polynomial Division with Remainder

F= x2 y+xy2 + y2,g= y2 -1,h=xy-1• F=(x+1)g+xh+2x+1• F=(x+y)h+g+x+y+1

• The remainders are different!• Can we somehow have a determinate division

algorithm, just like the case of Euclidean algorithm and Gaussian elimination? How to define a remainder?

Thus the task is, not so much to see what no one has yet seen; but to think what no one has yet thought about that which everyone sees. -- Erwin Schrödinger, Austrian physicist, (1887--1961)

Term OrdersA term order is a total order < on Nn, or the set of all monomials xa = x1

a1x2a2 ··· xn

an such that:(1) it is compatible with multiplication:

xa < xb ⇒ xa+c < xb+c

(2) it is a well-ordering, namely every nonempty subset of Nn has a smallest element under the term order.

N: Nonnegative integers

Example Term Orders• In one variable, there is only one

term order: 1 < x < x2 < x3 < ···• For n = 2, we have

– degree lexicographic order (deglex)1 < x1 < x2 < x1

2 < x1x2 < x22 < x1

3 < x12x2 < ···

– purely lexicographic order (lex)1 < x1 < x1

2 < x13 < ··· < x2 < x1x2 < x1

2x2 < ···

Quiz: Show that there are infinitely many term orders on N2.

Initial Ideal• Every polynomial f ∈ K[x1,…,xn] has an

initial monomial, denoted by in<(f).• For every ideal I of K[x1,…xn] the

initial ideal of I is generated by all initial monomials of polynomials in I:

in<(I) = < in<(f) | f is in I >

Remainder Modulo an IdealFor any ideal I and polynomial f of K[x1,…,xn], the remainder of f modulo I w.r.t. given term order < is the uniqueelement r such that in<(r) does not lie in in<(I) and f-r lies in I.

f=g+r=g’+r’, g,g’ ∈ I => in<( r-r’) ∈ in<(I) => r=r’

This remainder r can be thought of as a normal form of f modulo I w.r.t. <.

Defining Gröbner BasesA finite subset G of an ideal I is a Gröbner basis(with respect to the term order <) if

{ in<(g) | g is in G }generates in<(I).

Note: This condition imposed on G guarantees the division algorithm on G terminate in finite steps and that the remainder of any polynomial f modulo G is just the normal form of f modulo I w.r.t. <.

Gröbner basis of I is a generating set of I and there are many such generating sets. For instance, we can add any element of I to G to get another Gröbner basis.

Standard Monomials

• I ⊂ Q[x1,…,xn] an ideal, < a term order. A monomial xa = x1

a1x2a2 ··· xn

an is standard if it is not in the initial ideal in<(I).

• Example: If n = 3 and in<(I) = <x13,x2

4,x35>,

the number of standard monomials is 60. If in<(I) = <x1

3,x24, x1x3

4>, then the number of standard monomials is infinite.

The Residue Ring• Theorem: The set of standard monomials is

a Q-basis for the residue ring Q[x1,…,xn]/I. I.e., modulo the ideal I, every polynomial f can be written uniquely as a Q-linear combination of standard monomials.

• Residue Ring ⇔ Normal Forms of Polynomials

• Quiz: For any V ⊆ Qn, its vanishing ideal is I(V)={f∈ Q[x1,…,xn]: f(x)=0, ∀x ∈ V}. Show that﹟{standard monomials of I(V)}= ﹟V.

Reduced Gröbner Bases• A reduced Gröbner basis satisfies:

(1) For each g in G, the coeff of in<(g) is 1;(2) The set { in<(g) | g is in G } minimally generates in<(I) (nothing can be removed);

(3) No trailing term of any g in G lies in the initial ideal in<(I).

• Theorem: Fixing an ideal I in K[x1,…,xn] and a term order <, there is a unique reduced Gröbner basis for I.

Dickson’s Lemma 定理： Nd中任一集合在按照逐分量比较大小的偏序关系下只有有限多个极小元素。

证明： 对维数d归纳。

Existence of Gröbner basis follows directly from Dickson’s Lemma !

Dickson, L. E. (1913). "Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors". Amer. Journal Math. 35: 413–422.

Term Orders (Cont’d)Another useful consequence of Dickson’s Lemma is that a term ordercan also be defined as a total order < on the set of all monomials xa = x1

a1x2a2 ··· xn

an such that:(1) it is compatible with multiplication:

xa < xb ⇒ xa+c < xb+c

(2’) the constant monomial is smallest:1 < xa for all a in Nn\{0}

Hilbert Basis TheoremTheorem: Every ideal in the polynomial ring K[x1,…,xn] is finitely generated.

This means that any ideal I has the form <F> for a finite set F of polynomials.– Note: for the 1-variable ring K[x1], every ideal

I is principal; that is, I is generated by 1 polynomial. This is the Euclidean Algorithm.

– Hilbert Basis Theorem follows from Dickson’s lemma!

Hilbert's first work was on invariant theory and, in 1888, he proved his famous Basis Theorem. Twenty years earlier Gordan had proved the finite basis theorem for binary forms using a highly computational approach. Attempts to generalise Gordan's work to systems with more than two variables failed since the computational difficulties were too great. Hilbert himself tried at first to follow Gordan'sapproach but soon realised that a new line of attack was necessary. He discovered a completely new approach which proved the finite basis theorem for any number of variables but in an entirely abstract way. Although he proved that a finite basis existed his methods did not construct such a basis.

http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Hilbert.html

Hilbert (1862--1943)

Hilbert in 1885

Noetherian PropertyIt was specifically for the proof of this theorem that Hilbert introduced the idea of a Noetherianring, and proved the Noetherian property of the polynomial ring (although this may sound absurd, since Emmy Noether, in whose honour the term Noetherian was subsequently introduced, was still a baby at the time Hilbert published his work on invariant theory.) -- I.R. Shafarevich, Basic Notions of Algebra, 科学出版社， 2006.

Amalie Emmy NoetherNoether's mathematical work has been divided into three "epochs". In the first (1908–1919), she made significant contributions to the theories of algebraic invariants and number fields. Her work on differential invariants in the calculus of variations, Noether's theorem, has been called "one of the most important mathematical theorems ever proved in guiding the development of modern physics". In the second epoch, (1920–1926), she began work that "changed the face of [abstract] algebra".…Noether developed the theory of ideals in commutative rings into a powerful tool with wide-ranging applications. She made elegant use of the ascending chain condition, and objects satisfying it are named Noetherian in her honor. In the third epoch, (1927–1935), she published major works on noncommutative algebras and hypercomplex numbers and united the representation theoryof groups with the theory of modules and ideals.

http://en.wikipedia.org/wiki/Emmy_Noether

Emmy Noether1882–-1935

• Dickson 1913 Hilbert 1890 • Noether 1920 Hilbert 1890

关于Noether给出的几个初等的Hilbert Basis Theorem的证明，包括构造性证明，请参见：

万哲先，有限反射群的不变式论，离散数学丛书，上海交大出版社，1998.

Hilbert的有限生成定理和他的基定理刚

刚出现的时候，有的数学家认为是神学，也有数学家认为Hilbert结束了不变式论的生命。 这都是不正确的。 －－万哲先，

有限反射群的不变式论，离散数学丛书，上海交大出版社，1998.

Felix Klein: ErlangenProgram 1872

With every geometry, Klein associated an underlying group of symmetries. The hierarchy of geometries is thus mathematically represented as a hierarchy of these groups, and hierarchy of their invariants.

http://en.wikipedia.org/wiki/Erlangen_program

Paul Albert Gordan, a mathematician at the University of Erlangen. Emmy Noether was Gordan's only doctoral student.

An often repeated story is that Gordan's reponse to Hilbert's proof was the exclamation, "Das ist nicht Mathematik, das ist Theologie!." Weyl's comment is likewise memorable: "What then would he have said about his former pupil Emmy Noether's later "theology", that abhorred all calculation and operated in a much thinner air of abstraction than Hilbert ever dared!" -- http://faculty.evansville.edu/ck6/bstud/gordan.html

Paul Albert Gordan (1837 – 1912)

Carl Jacobi Paul Gordan Emmy NoetherWolfgang Gröbner Bruno Buchberger Bernd Sturmfels

Like the Arabian phoenix rising out of its ashes, the theory of invariants, pronounced dead at the turn of the century, is once again at the forefront of mathematics. During its long eclipse, the language of modern algebra was developed, a sharp tool now at last being applied to the very purpose for which it was invented. – J.P. Kung, G.-C. Rota, The invariant theory of binary forms, Bull. Amer. Math. Soc. 10 (1984) 27—85.

The quote refers to the fact that three of Hilbert’s fundamental contributions to modern algebra, namely, the Nullstellensatz, the Basis Theorem and the Syzygy Theorem, were first proved as lemmas in his invariant theory papers(Hilbert 1890, 1893). It is also noteworthy that, contrary to a common belief, Hilbert’s main results in invariant theory yield an explicit finite algorithm for computing a fundamental set of invariants for all classical groups. – B. Sturmfels, Algorithms in Invariant Theory, Springer, 2008.

Early History of Algebraic Invariant Theory

• 1801, Gauss’s work on quadratic binary form• 1841, Boole published a book on invariant

theory and really launched a study of it. (P.R. Wolfson, George Boole and the origins of invariant theory, Historia Mathematica,35, (2008), 37-46)

• British School: Boole (famous for Boolean algebra), Cayley, Sylvester, Salmon…

• German School: Hesse (famous for Hessian), Aronhold, Gordan, Clebsch…

Algebraic GeometryIf F is a set of polynomials, the variety of F over the complex numbers C equals

V(F) = {(z1,…,zn) ∈ Cn | f(z1,…,zn) = 0, ∀f ∈ F}

Note: The variety depends only on the ideal of F. I.e. V(F) = V(<F>). If G is a Gröbner Basis for F, then V(G) = V(F).

Fundamental Building BlocksThe basis theorem lies at the very foundations of algebraic geometry; it shows there are ``fundamental building blocks,’’ in the sense that each variety is uniquely the finite union of irreducible varieties. This is very much akin to the fundamental theorem of arithmetic, which lies at the foundations of number theory; it says that every integer is a product of primes (the ``building blocks’’), and that this representation is unique (up to order and units). -- K. Kendig, Elementary Algebraic Geometry, p. 119, Springer, 1977.

Hilbert’s Nullstellensatz• Theorem (David Hilbert, 1890):

V(F) is empty if and only if G = {1}.– Easy direction: if G = {1}, then V(F) = V(G) = { }

• Ex: F = {x2+xy-10, x3+xy2-25, x4+xy3-70}. Here, G = {1}, so there are no common solutions. Replacing 25 above by 26, we have G = {x-2,y-3} and V(F) = V(G) = {(2,3)}.

Fundamental Theorem of Algebra

• Theorem: The number of standard monomials equals #V(I), where the zeroes are counted with multiplicity.

• Example: F = {x2z-y, x2+xy-yz, xz2+xz-x}. Then, using purely lex order x > y > z, we get G = { x2-yz-y, xy+y, xz2+xz-x, yz2+yz-y, y2-yz }. Every power of z is standard, so #V(F) is infinite.

• Replacing x2z-y with x2z-1 in F, we get G = { x-2yz+2y+z, y2+yz+y-z-3/2, z2+z-1 } so that #V(F) = 4.

Generalization of Hilbert’s Nullstellensatz

Dimension of a Variety• Calculating the dimension of a variety

– Think of dimension intuitively: points have dimension 0, curves have dimension 1, ….

• Let S ⊂ {x1,…,xn} have maximal cardinality with the property that no monomial in the variables in S appears in in<(I).

• Theorem: dim V(I) = #S

Determinantal Varieties

Subtraction-Polynomial• Take g,g’ in G and form the S-polynomial m’g - mg’

where m,m’ are monomials of lowest degree s.t. m’⋅in<(g) = m⋅in<(g’).

• Cancellation of leading terms among polynomials of the same multidegree can be accounted for by S-polynomials.

• The notion of S-polynomial is the nucleus of algorithmic Gröbner bases theory. Note, howeve, that the notion of Gröbner bases is independent of the notion of S-polynomials and gives many interesting results also for nonalgorithmic polynomial ideal theory.– B. Buchberger, Gröbner bases: A short introduction for systems theorists, 2001.

Buchberger’s CriterionTheorem (Buchberger’s Criterion): G is a Gröbner basis if and only if every S-polynomial formed by pairs g,g’ from G has normal form zero w.r.t. G.

This is the key result of Gröbner basis!

Buchberger’s Algorithm• Input: Finite list F of polynomials in Q[x1,…,xn]• Output: The reduced Gröbner basis G for <F>.• Step 1: Apply Buchberger’s Criterion to check whether F is

a Gröbner basis.• Step 2: If “yes,” then F is a GB. Go to Step 4.• Step 3: If “no,” we found p = normalf(m’g-mg’) to be nonzero.

Set F = F ∪ {p} and go to Step 1.• Step 4: Replace F by the reduced Gröbner basis G (apply

“autoreduction’’) and output G.

Termination of Algorithm• Question: Why does this loop always

terminate? Step 1 Step 3

• Answer: Hilbert’s Basis Theorem implies that there is no infinite ascending chain of ideals. Let F = {f1,…,fd}. Each nonzero p = normalf(m’f-mf’) gives a strict inclusion: <in<(f1),…,in<(fd)> ⊂<in<(f1),…,in<(fd), in<(p)> . Hence the loop terminates.

Simple ExampleExample: n = 1, F = {x2+3x-4,x3-5x+4}– form the S-poly (Step 1):

x(x2+3x-4) - 1(x3-5x+4) = 3x2+x-4 It has nonzero normal form p = -8x+8.– Therefore, F is not a Gröbner basis.

We enlarge F by adding p (Step 3).– The new set F ∪ {p} is a Gröbner basis.– The reduced GB is G = {x-1} (Step 4).

Syzygy (合冲)• Syzygy: the nearly straight-line

configuration of three celestial bodies in a gravitational system.

• (h1,…,hd) is a syzygy among f1,…,fd :h1 f1+…+ hd fd =0

• Just as the sun or moon is obscured during an eclipse, leading terms of polynomials are obscured by syzygies.

Syzygy (Cont’d)The notion of syzygy was introduced already by Boole. The basic problems since then on invariant theory include:

• Determine a minimal set of invariants so that all others are obtained using polynomials of them.

• Determine a minimal set of syzygies for given invariants.

Hilbert's Syzygy TheoremIn mathematics, Hilbert's syzygy theorem is a result of commutative algebra, first proved by David Hilbert (1890) in connection with the syzygy(relation) problem of invariant theory. Roughly speaking, starting with relations between, then relations between the relations, and so on, it explains how far one has to go to reach a clarified situation. It is now considered to be an early result of homological algebra, and through the depth concept, to be a measure of the non-singularity of affine space.

http://en.wikipedia.org/wiki/Hilbert's_syzygy_theorem

Syzygy Theorems• Generalized Hilbert Basis Theorem: The

module of syzygies among a set of polynomials is finitely generated.

• Extension of Hilbert Basis Theorem and Buchberger algorithm to syzygies was done in the PhD thesis of Schreyer in 1980 and is reported here:

• D. Eisenbud, Commutative Algebra: With a View Toward Algebraic Geometry, Theorem 15.10, Springer, 1995.

Grassmann-Plücker Syzygy

We now derive the Grassmann-Plückeridentity, which would be the most important identity for determinants in the context of the combinatorial study of matrices.

--- K. Murota, Matrices and Matroids for Systems Analysis, p. 34, Springer, 2000.

Grassmann-Plücker IdentityLet A be a matrix with |R| ≤|C| where R=Row(A) and C=Col(A). For two subsets J and J’ of C with |J|=|J’|=|R|, it holds

det A[R,J]·det A[R,J’]=∑det A[R,J-i+j]·detA[R,J’+i-j]

where the summing index runs through all elements j of J’﹨J and J-i+j is a shorthand for the column arrangement obtained from J by replacing column i with column j.

Quiz: Show that three-term Grassmann-Plücker identities generate all Grassmann-Plückeridentities.

• In summary, Gröbner bases and the BuchbergerAlgorithm for finding them are fundamental notions in algebra. They furnish the engine for more advanced computations in algebraic geometry, such as elimination theory, computing cohomology, resolving singularities, etc.

• Given that polynomial models are ubiquitous across the sciences and engineering, Gröbner bases have been used by researchers in optimization, coding, robotics, control theory, statistics, molecular biology, and many other fields.

• We invite the reader to experiment with one of the many implementations of Buchberger’salgorithm (e.g., in CoCoA, Macaulay2, Magma, Maple, Mathematica, or Singular).

Suggested Literature• B. Sturmfels, Algorithms in Invariant

Theory, Springer, 2008.• B. Hassett, Introduction to Algebraic

Geometry, Cambridge University Press, 2007.

• R.R. Thomas, Lectures in Geometric Combinatorics, AMS, 2006.

• K. Murota, Matrices and Matroids for Systems Analysis, Springer, 2000.

• 刘木兰， Gröbner基理论及其应用，科学出版社，2000。

• 万哲先，有限反射群的不变式论，离散数学丛书，上海交大出版社，1998.

Suggested Literature

• D. Cox, J. Little, D. O’Shea, Ideals, Varieties, and Algorithms, 世界图书出版公司, 2004.

• N. Lauritzen, Concrete Abstract Algebra: From Numbers to GröbnerBases, Cambridge University Press, 2003.

• D. Cox, J. Little, D. O’Shea, Using Algebraic Geometry, Springer, 1998.

Suggested Literature

• Victor V. Prasolov, Polynomials, Springer, 2004.

• B. Buchberger, F. Winkler (eds.), GröbnerBases and Applications,Cambridge University Press, 1998.

谢谢！